Solving Integrals with Exercise Sheet, Exercises of Calculus

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@ Exercise #5: Use separate sheet/s (short bond paper) for your solutions and answers. Solve the following integrals. sin? xdx 1. fsin 5x cos 3x dx il ee 3 2. fsin? x cos’ xdx jes 3. fcos*x sin? xdx 8. f sin? x tan x dx 4. fcos?x sin? xdx 9. [ cos* x cotx dx 5. f cos® 3x dx 10. f cot 2x dx Solutions and Answers 1. f sin 5x cos3xdx Use the identity: 5 [sin(m +n) +sin(m—-n)] a. 1 = 5 | sin + 3x) + sin(Sx — 3x) dx = 3! sin(8x) + sin(2x) dx 1 1 =5 | sin(8x) dx + 3! sin(2x) dx = 5 [g(-cosex)| +55 (-cos2x)] 2. fsin?x cos>x dx = | sin? x cos3xcos?xdx = J sine x cos3(1 — sin?x)dx sin*xcosx (1 — sin?x) — | sin*xcosx (1 — sin?x) = | sin? x cos*dx — | sin*x cos?x dx = | sin?xcosx dx — | sin*x cosx dx — | sin*x cosx dx + i sin®x cosx dx